Cellular Automata: From Structural Principles to Transport and Correlation Methods

TL;DR

This survey explores cellular automata's structural principles, transport phenomena, and correlation methods, linking microscopic rules to macroscopic behaviors in statistical physics.

Contribution

It provides a unified view of CA as shift-commuting maps, classifies transport regimes, and develops correlation-based tools for analysis and coarse-graining.

Findings

CA support diverse phenomena like phase transitions and criticality.

Transport in CA can be categorized into ballistic, diffusive, and anomalous regimes.

Correlation methods enable regime diagnosis and coarse-graining.

Abstract

Cellular automata (CA) are discrete-time dynamical systems with local update rules on a lattice. Despite their elementary definition, CA support a wide spectrum of macroscopic phenomena central to statistical physics: equilibrium and nonequilibrium phase transitions, transport and hydrodynamic limits, kinetic roughening, self-organized criticality, and complex spatiotemporal correlations. This survey focuses on three tightly connected themes. \emph{(i)} We present a structural view of CA as shift-commuting maps on configuration spaces, emphasizing rule complexity, reversibility, and conservation laws (including discrete continuity equations). \emph{(ii)} We organize transport in CA into ballistic, diffusive, and anomalous regimes, and connect microscopic currents to macroscopic laws through Green--Kubo formulas, scaling theory, and universality classes. \emph{(iii)} We develop correlation-based methods -- from structure factors and response formulas to computational mechanics and data-driven inference -- that diagnose regimes and enable coarse-graining.